Algebra 2 Honors: Topic 1

1-1 Key Features of Functions

How to Find the Domain of a Function

Rules:

  1. No division by 0, aka NO ZERO IN DENOMINATOR

  2. Cannot take the sq root of a negative number

Ex: p(x)=4x−5p\left(x\right)=\sqrt{\frac{4}{x-5}}

  1. We define what we want to get the domain.

    1. Can’t have 0 in denominator, so x−5≠0\sqrt{x-5}\ne0

    2. Can’t take sq root of negative, so x−5≥0x-5\ge0

  2. We solve for x to see what numbers it can be

    1. x > 5

  3. Write the domain

    1. Dp = {x | x > 5}

9-1 Symmetry

Types of Symmetry:

Symmetry along the y-axis: (x,y) → (-x,y)

If replacing the x in the equation with -x keeps the equation the same, it’s symmetrical.

Symmetry along the x-axis: (x,y) → (x,-y)

If replacing the y in the equation with -y keeps the equation the same, it’s symmetrical.

Symmetry to the origin: (x,y) → (-x,-y)

If replacing the x and y in the equation with -x and -y keeps the equation the same, it’s symmetrical.

Even and Odd Functions

Even: Functions where f(-x) = f(x) (also symmetry to the y-axis)

Odd: Functions where f(-x) = -f(x) (also symmetry to the origin)

1-2b Dilations with Functions

How to do a translation:

Ex: 12f(x−3)−2\frac12f\left(x-3\right)-2

  • ALWAYS start from the inside then go outword, while following order of operations

Rules:

For (x-a): Move RIGHT a times

For (x+a): Move LEFT a times

For (x ___) + b: Move UP b times

For (x ___) - b: Move DOWN b times

For A⋅f(x)A\cdot f\left(x\right) : Graph goes through VERTICAL STRETCH (A>1)

For 1A⋅f(x)\frac{1}{A}\cdot f\left(x\right) : Graph goes through VERTICAL COMPRESSION (0 < A < 1)

For f(Bx)f\left(Bx\right) : Graph goes through HORIZONTAL COMPRESSION (you divide x/b)

For f(1B⋅x)f\left(\frac{1}{B}\cdot x\right) : Graph goes through HORIZONTAL STRETCH (you multiply the x by b)

IF A < 0, it reflects over the x-axis

IF B < 0, it reflects over the y-axis

1-3 Piecewise Defined Functions

  • They’re functions with multiple possibilities, but they DON’T OVERLAP

  • Ex: When doing Vertical Line Test, it passes with all possibilities graphed together